Partitioning Point Sets in 4 Dimensions
Partitioning Point Sets in 4 Dimensions
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(in our construction 8 = 1/32, so that m = \/lOO suffices. ) Call this the continuous set.
Lemma 5: If there is a 8(n+m)-partition Q' of the continuous set then there b a fin-partition of the point set S.
Proof: Qearly a 8(n+m)-paitition of the continuous set implies a [8n-(l— 8)in]-partition of the sphere set, and hence of the point set 5. On a point set diis is a (f8n - (l-8)ml)-partition, which by assumption is a 8n-partition. □ Thus to obtain our result it suffices to show that a body S, of... mass n, has an /t/32-partition. We use the following two lonmas in our constnictioa.
Lemom 6: [W] Given a connected body S of mass n, in two dimensional Eudidean space, and given a line L forming an n/2-partition of S, there is a unique line K such that L and K form an n/4 partition. Furthermore, K varies continuously with L.
^^ard did not prove exactly this theoron, but the proof would be identical to Willard's. The fact that 5 is connected guarantees uniqueness.
Lemma 7: (Borsuk-Ulum theorem [L]).
What to read after Partitioning Point Sets in 4 Dimensions?
You can find similar books in the "Read Also" column, or choose other free books by Richard Cole to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
(in our construction 8 = 1/32, so that m = \/lOO suffices. ) Call this the continuous set.
Lemma 5: If there is a 8(n+m)-partition Q' of the continuous set then there b a fin-partition of the point set S.
Proof: Qearly a 8(n+m)-paitition of the continuous set implies a [8n-(l— 8)in]-partition of the sphere set, and hence of the point set 5. On a point set diis is a (f8n - (l-8)ml)-partition, which by assumption is a 8n-partition. □ Thus to obtain our result it suffices to show that a body S, of... mass n, has an /t/32-partition. We use the following two lonmas in our constnictioa.
Lemom 6: [W] Given a connected body S of mass n, in two dimensional Eudidean space, and given a line L forming an n/2-partition of S, there is a unique line K such that L and K form an n/4 partition. Furthermore, K varies continuously with L.
^^ard did not prove exactly this theoron, but the proof would be identical to Willard's. The fact that 5 is connected guarantees uniqueness.
Lemma 7: (Borsuk-Ulum theorem [L]).
What to read after Partitioning Point Sets in 4 Dimensions?
You can find similar books in the "Read Also" column, or choose other free books by Richard Cole to read onlineMoreLess
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